{
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   "source": [
    "Chapter 02\n",
    "\n",
    "# 向量积\n",
    "Book_4《矩阵力量》 | 鸢尾花书：从加减乘除到机器学习 (第二版)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "873da06f-c90c-4a2a-abd3-7f8711feca02",
   "metadata": {},
   "source": [
    "此代码定义了两个三维向量 $a$ 和 $b$，并计算了它们的叉积（向量积）。首先定义了行向量形式的 $a$ 和 $b$，然后分别计算了行向量和列向量的叉积。\n",
    "\n",
    "### 叉积公式\n",
    "对于三维向量 $a = \\begin{bmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{bmatrix}$ 和 $b = \\begin{bmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{bmatrix}$，叉积定义为：\n",
    "\n",
    "$$\n",
    "a \\times b = \\begin{bmatrix} a_2 b_3 - a_3 b_2 \\\\ a_3 b_1 - a_1 b_3 \\\\ a_1 b_2 - a_2 b_1 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "代码中计算了行向量和列向量形式的叉积。结果表示两个向量的垂直方向，并可用于三维空间中的法向量计算。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e369506-697f-450e-a731-76945415ed9c",
   "metadata": {},
   "source": [
    "## 导入所需库"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "dbb6f14c-fcb4-499f-9618-d6ca9c8b36ab",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np  # 导入NumPy库，用于数值计算"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d9fa082-6e75-4a30-9ff1-b119c73a3a25",
   "metadata": {},
   "source": [
    "## 定义两个行向量"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "081b4fe5-be16-4854-914b-5c795f937fa8",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.array([-2, 1, 1])  # 定义向量a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "9a902aa8-799a-4461-a5c0-459f2550272a",
   "metadata": {},
   "outputs": [],
   "source": [
    "b = np.array([1, -2, -1])  # 定义向量b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b04a2e40-0ae4-44f1-a6bd-e691ee781d18",
   "metadata": {},
   "source": [
    "## 计算行向量的叉积"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "95ea922a-d94c-42c3-8459-d8fc755b8df3",
   "metadata": {},
   "outputs": [],
   "source": [
    "a_cross_b = np.cross(a, b)  # 计算a和b的叉积"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6de6f635-f9e9-4667-a12a-884ccaeb26f3",
   "metadata": {},
   "source": [
    "## 定义两个列向量"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "0d9f2b40-f6a0-4ca6-a70b-3910b69a9706",
   "metadata": {},
   "outputs": [],
   "source": [
    "a_col = np.array([[-2], [1], [1]])  # 定义列向量a_col"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "66e48d3a-7fa8-41a8-bf8b-61cfdf250200",
   "metadata": {},
   "outputs": [],
   "source": [
    "b_col = np.array([[1], [-2], [-1]])  # 定义列向量b_col"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c19ed8a3-f8e7-4c42-b6f7-bac01f31e016",
   "metadata": {},
   "source": [
    "## 计算列向量的叉积"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "9c019178-15f5-41f6-93d5-ceea5505684f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1],\n",
       "       [-1],\n",
       "       [ 3]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_cross_b_col = np.cross(a_col, b_col, axis=0)  # 计算a_col和b_col的叉积，沿axis=0进行计算\n",
    "a_cross_b_col"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "85a80909-2aac-49ed-bb7a-f8cc6b80ee7d",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ecd322f4-f919-4be2-adc3-69d28ef25e69",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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